1. Field of the Invention
The present invention relates to the field of signal coding, and more specifically to Karhunen-Loeve transform coding.
2. Discussion of the Background
Transform coding is a known technology for coding signals. FIG. 1 shows a block diagram of a conventional transform coder, including a transmitter 10 and a receiver 11. At the transmitter 10, an original discrete-time signal frame xn is transformed by a transformer 100 into transform coefficients yn. Each signal frame xn, for n=1, 2, 3, . . . is generally comprised of a continuous segment of length N of a discrete-time signal x(k) and can be defined, for example, as xn=[x(nN−1)x(nN−2) . . . x(N(n−1))]. Further, the segments of the signal x(k) defining the signal frames xn can overlap up to N−1 elements or be totally non-overlapping.
The transformation preferably decorrelates the original signal frame and compacts a large fraction of the signal energy, i.e. variance, into relatively fewer transform coefficients. This is known as an energy-packing property. Because of this property, it is possible to retain a fraction of the transform coefficients without seriously affecting the reconstructed signal quality. The transform coefficients are quantized at the quantizer 110 to yield the quantized transform coefficients ŷn. As a result of quantization and the possible elimination of redundant transform coefficients, the quantized signal ŷn that is encoded by the encoder 120 may contain significantly less than the N elements in the original signal frame xn. The encoded signal is then sent along the channel.
At the receiver 11, the encoded signal is received through the channel into the decoder 130. The signal is decoded, yielding the quantized transform coefficients ŷn. The original signal frame is reconstructed at the inverse transformer 140 to produce the reconstructed signal frame {circumflex over (x)}n. Because the transform coefficients were quantized and some redundant or insignificant transform coefficients were possibly eliminated prior to encoding, the reconstructed signal frame {circumflex over (x)}n is generally slightly different from the original signal frame xn.
The Karhunen-Loeve transform (KLT) is known to be the optimum transform for signal compression because the KLT exhibits a significant energy-packing property. In other words, a larger fraction of the total energy or variance of the signal frame is contained in the first few coefficients, as compared to other transforms. Therefore, the number of transform coefficients retained, quantized, and encoded can be significantly less than other transforms. As a result, the signal bit rate during transmission is reduced while the signal quality is maintained. More details concerning the KLT are found in Digital Coding of Waveforms, by N. Jayant and P. Noll, 1984, the entire contents thereof incorporated herein by reference. Additional information about KLT may also be found in (1) C. E. Davila, “Blind Adaptive Estimation of KLT Basis Vectors”, Transactions on Signal Processing, Vol. 49, No. 7, pgs. 1364–1369, July, 2001; (2) C. E. Davila, “Blind KLT Coding,” Proc. 2000 IEEE Signal Processing Workshop, Hunt, Tex., Oct. 15–18, 2000 (the proceedings of which are available via HTTP at “spib.rice.edu/SPTM/DSP2000/submission/DSP/papers/paper153/paper153.pdf”); and (3) C. E. Davila, “Blind Adaptive Estimation of KLT Basis Vectors,” Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing, Istanbul, Jun. 5–9, 2000. The contents of those papers are incorporated herein by reference.
In the transformer 100, the KLT transform coefficients yn are determined by applying a KLT transform matrix Q to the original signal xn, as shown in Equation (1),yn=Qxn.  (1)The net effect of the transformation is to establish a new coordinate system whose origin is at the centroid of the population of the original signal frame xn and whose axes are in the direction of the eigenvectors of the autocorrelation matrix R of the original signal frame xn. The eigenvalues are the variances, i.e., original signal energy, of the KLT transform coefficients yn along the eigenvectors. The eigenvectors form the column vectors of the transform matrix Q, and are also referred to as KLT basis vectors.
Unfortunately, the KLT basis vectors are data-dependent, i.e., they require an estimate of the autocorrelation matrix of the original signal frame xn for their computation. Hence, in order to reconstruct the original signal frame at the receiver 11, the KLT basis vectors must also be encoded with the KLT transform coefficients and transmitted to the receiver. The necessity of transmitting the KLT basis vectors reduces signal compression and leads to increased bit rates. For this reason, the KLT has had limited use in signal coding algorithms.